# Two's Complement: Integer -> Binary: 2 684 354 603 Convert the Integer Number to a Signed Binary in Two's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

## Signed integer number 2 684 354 603(10) converted and written as a signed binary in two's complement representation (base 2) = ?

### 1. Divide the number repeatedly by 2:

#### We stop when we get a quotient that is equal to zero.

• division = quotient + remainder;
• 2 684 354 603 ÷ 2 = 1 342 177 301 + 1;
• 1 342 177 301 ÷ 2 = 671 088 650 + 1;
• 671 088 650 ÷ 2 = 335 544 325 + 0;
• 335 544 325 ÷ 2 = 167 772 162 + 1;
• 167 772 162 ÷ 2 = 83 886 081 + 0;
• 83 886 081 ÷ 2 = 41 943 040 + 1;
• 41 943 040 ÷ 2 = 20 971 520 + 0;
• 20 971 520 ÷ 2 = 10 485 760 + 0;
• 10 485 760 ÷ 2 = 5 242 880 + 0;
• 5 242 880 ÷ 2 = 2 621 440 + 0;
• 2 621 440 ÷ 2 = 1 310 720 + 0;
• 1 310 720 ÷ 2 = 655 360 + 0;
• 655 360 ÷ 2 = 327 680 + 0;
• 327 680 ÷ 2 = 163 840 + 0;
• 163 840 ÷ 2 = 81 920 + 0;
• 81 920 ÷ 2 = 40 960 + 0;
• 40 960 ÷ 2 = 20 480 + 0;
• 20 480 ÷ 2 = 10 240 + 0;
• 10 240 ÷ 2 = 5 120 + 0;
• 5 120 ÷ 2 = 2 560 + 0;
• 2 560 ÷ 2 = 1 280 + 0;
• 1 280 ÷ 2 = 640 + 0;
• 640 ÷ 2 = 320 + 0;
• 320 ÷ 2 = 160 + 0;
• 160 ÷ 2 = 80 + 0;
• 80 ÷ 2 = 40 + 0;
• 40 ÷ 2 = 20 + 0;
• 20 ÷ 2 = 10 + 0;
• 10 ÷ 2 = 5 + 0;
• 5 ÷ 2 = 2 + 1;
• 2 ÷ 2 = 1 + 0;
• 1 ÷ 2 = 0 + 1;

## How to convert signed integers from decimal system to signed binary in two's complement representation

### Follow the steps below to convert a signed base 10 integer number to signed binary in two's complement representation:

• 1. If the number to be converted is negative, start with the positive version of the number.
• 2. Divide repeatedly by 2 the positive representation of the integer number, keeping track of each remainder, until we get a quotient that is zero.
• 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
• 4. Binary numbers represented in computer language must have 4, 8, 16, 32, 64, ... bit length (a power of 2) - if needed, add extra bits on 0 in front (to the left) of the base 2 number above, up to the required length, so that the first bit (the leftmost) will be 0, correctly representing a positive number.
• 5. To get the negative integer number representation in signed binary one's complement, replace all 0 bits with 1s and all 1 bits with 0s (reversing the digits).
• 6. To get the negative integer number, in signed binary two's complement representation, add 1 to the number above.

### Example: convert the negative number -60 from the decimal system (base ten) to signed binary in two's complement:

• 1. Start with the positive version of the number: |-60| = 60
• 2. Divide repeatedly 60 by 2, keeping track of each remainder:
• division = quotient + remainder
• 60 ÷ 2 = 30 + 0
• 30 ÷ 2 = 15 + 0
• 15 ÷ 2 = 7 + 1
• 7 ÷ 2 = 3 + 1
• 3 ÷ 2 = 1 + 1
• 1 ÷ 2 = 0 + 1
• 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above:
60(10) = 11 1100(2)
• 4. Bit length of base 2 representation number is 6, so the positive binary computer representation of a signed binary will take in this particular case 8 bits (the least power of 2 larger than 6) - add extra 0 digits in front of the base 2 number, up to the required length:
60(10) = 0011 1100(2)
• 5. To get the negative integer number representation in signed binary one's complement, replace all the 0 bits with 1s and all 1 bits with 0s (reversing the digits):
!(0011 1100) = 1100 0011
• 6. To get the negative integer number, signed binary in two's complement representation, add 1 to the number above:
-60(10) = 1100 0011 + 1 = 1100 0100